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Last summer there was a long thread with the subject "My first Lunar":
(http://groups.google.com/group/NavList/browse_thread/thread/3d6cb12c5f97d5ba?hl=en#)

I'm half a year late, but here are my results:

time:
1999 Jan 26 22:19:32.090 UTC
position:
N 14.56993° W 61.66016°

To test those results, I have azimuths and altitudes (from the JPL
HORIZONS online calculator) for that time and place. If my solution is
correct, the Moon and Jupiter altitudes and the lunar distance should
match the observations. (In order to minimize roundoff error in this
evaluation, I've used greater precision than is normally reasonable.)

HORIZONS says the Moon azimuth is 83.2772° and refracted altitude is
65.7533°. Compare that to the altitude observation:

  66.15500  Moon upper limb, sextant
    .12833  dip
    .27388  semidiameter (HORIZONS value)
  -------
  65.75279  observed refracted center altitude
  65.7533   HORIZONS altitude
  --------
   -.00051  intercept


HORIZONS says Jupiter azimuth is 250.9957° and refracted altitude is
45.2444°. Compare that to the altitude observation:

  45.37167  Jupiter, sextant
    .12833  dip
  -------
  45.24333  observed refracted center altitude
  45.2444   HORIZONS altitude
  --------
   -.00107  intercept

The observed altitudes are 2 and 4 arc seconds different from the
HORIZONS predictions.


 From the HORIZONS azimuths and altitudes I calculate 68.59563°
refracted lunar distance, center to center. Compare that to the observation:

  68.32333  sextant, near limb to center
    .27388  Moon semidiameter (HORIZONS value)
  --------
  68.59721  observed center to center refracted angle
  68.59563  HORIZONS
  --------
   -.00158  difference

The observed lunar distance is 6 arc seconds different from the HORIZONS
value.

In terms of practical sextant accuracy, the solution is almost perfect.
That is, when you give HORIZONS the time and position I computed, its
Moon and Jupiter positions match all three observed angles within a
tenth of an arc minute.

Here is the full output of my program, in units and precision more
appropriate to navigation:


Program Lunar2, by Paul S. Hirose.

Initial conditions.

estimated time:
1999-01-26T22:20:00.0 UTC
1999-01-26T22:21:04.0 Terrestrial Time
63.492 seconds delta T

estimated position:
+20°00.0' - 70°00.0' north lat, east lon
           - 70°15.9' ephemeris east lon
19 meters above ellipsoid

atmosphere:
15° C (59° F) at observer
1013.3 mb (29.92" Hg) altimeter setting
1011.0 mb (29.85" Hg) actual pressure

Moon altitude observation:
  66°01.6' observed upper limb altitude
      0.4' refraction
     16.4' unrefracted semidiameter
  65°44.8' unrefracted altitude of center
  57°57.9' predicted altitude
   7°46.8' intercept
  91°49.7' predicted azimuth

Jupiter altitude observation:
  45°14.6' observed center altitude
      0.9' refraction
  45°13.7' unrefracted altitude of center
  50°06.9' predicted altitude
- 4°53.2' intercept
239°30.1' predicted azimuth

Moon to Jupiter predicted separation angle:
   68°42.7' center to center, unrefracted
       1.3' refraction
   68°41.4' center to center, refracted
      16.4' Moon near limb refracted semidiameter
   68°25.0' Moon near limb to Jupiter center
   68°19.4' observed angle
-  0°05.6' observed - predicted

separation angle rate of change:
+24" per minute (topocentric)
94% of total angular velocity

--------------------

Solution, after 5 iterations.

corrected time:
1999-01-26T22:19:32.0 UTC
1999-01-26T22:20:36.4 Terrestrial Time
63.492 seconds delta T

corrected position:
+14°34.2' - 61°39.6' north lat, east lon
           - 61°55.5' ephemeris east lon
12° LOP crossing angle

geocentric coordinates (true equator and equinox):
  4h14.72m +16°00'48"  Moon RA and dec.
16.2' apparent semidiameter
23h48.75m - 2°29'12"  Jupiter RA and dec.
0.3' semidiameter

geocentric separation angle and rate:
   68°13.1' center to center
+35" per minute
97% of total angular velocity

illumination conditions:
251.9° -4.7° Sun unrefracted az, el
347° Moon to Sun position angle (0 = 12 o'clock)
61° Moon phase angle (0 = full, 180 = new)
181° Jupiter to Sun position angle
9° Jupiter phase angle

position angles:
351° Moon to Jupiter
5° Jupiter to Moon

recommended limbs:
Use Moon upper limb.
Use Jupiter lower limb.
Use Moon near limb.
Use Jupiter far limb.

Moon altitude observation:
  66°01.6' observed upper limb altitude
      0.4' refraction
     16.4' unrefracted semidiameter
  65°44.7' unrefracted altitude of center
  65°44.7' predicted altitude
   0°00.0' intercept
  83°16.6' predicted azimuth

Jupiter altitude observation:
  45°14.6' observed center altitude
      0.9' refraction
  45°13.7' unrefracted altitude of center
  45°13.7' predicted altitude
   0°00.0' intercept
250°59.7' predicted azimuth

Moon to Jupiter predicted separation angle:
   68°37.2' center to center, unrefracted
       1.4' refraction
   68°35.8' center to center, refracted
      16.4' Moon near limb refracted semidiameter
   68°19.4' Moon near limb to Jupiter center
   68°19.4' observed angle
    0°00.0' observed - predicted

separation angle rate of change:
+22" per minute (topocentric)
93% of total angular velocity



With large errors in the initial estimates for time and position and the
narrow angle between the LOPs, this problem was a good challenge for the
computer. The program needed five iterations to reach a solution. For
the comparison with HORIZONS, I requested .00001° accuracy, causing the
program to iterate six times.

Its algorithm belongs to today's era of cheap computing power, when
iterating to the desired accuracy can be more practical than a direct
solution which demands less from the machine but more from the
programmer. There's no attempt to reduce the lunar to the equivalent
geocentric observation. Instead, the algorithm works directly with the
three topocentric observables (one separation angle and two altitudes)
to deduce the three unknowns (time, latitude, and longitude).

The assumptions are 1) you can predict the three observables, given a
set of values assigned to the unknowns, 2) fairly good initial estimates
of the unknowns are at hand, 3) a small change in any unknown (the other
two remaining constant) causes a proportional change in the three
predicted observables.

This implies that you can describe the response of the observables with
three linear equations in the three unknowns. Solving the equations
yields corrections to the initial estimates of the unknowns. That's the
end of the problem, if the relationship between the observables and
unknowns is truly linear. In reality, it's not, so what you get are
improved estimates of the unknowns. By repeating this process the lunar
can be solved to arbitrary accuracy.

That's the principle of my program. The source code is available my site:
http://home.earthlink.net/~s543t-24dst/sofajplNet/LunarDist2.html

In a followup message I'll try to explain the algorithm in more detail.

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